Solve Equation $\sqrt[3]{a-1}+\sqrt[3]{a}+\sqrt[3]{a+1}=0$ in Real Numbers

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Discussion Overview

The discussion revolves around solving the equation $\sqrt[3]{a-1}+\sqrt[3]{a}+\sqrt[3]{a+1}=0$ in real numbers. Participants explore different methods and approaches to find solutions, including algebraic manipulations and potential pitfalls in calculations.

Discussion Character

  • Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant presents the equation and seeks solutions.
  • Another participant attempts to solve the equation using a method that leads to the equation $3a = 3\sqrt[3]{a(a^2-1}$, suggesting $a^3 = a(a^2-1)$ and concludes with $a=0$ as a solution.
  • A later reply acknowledges a calculation mistake in their previous method but reiterates the conclusion that $a=0$ is the only solution.
  • Another participant expresses appreciation for the contributions and confirms that $a=0$ is the only answer.

Areas of Agreement / Disagreement

While one participant initially presents a different answer, there is a subsequent acknowledgment of a calculation mistake, leading to a consensus that $a=0$ is the only solution. However, the initial disagreement on methods and answers indicates that multiple views were present during the discussion.

Contextual Notes

The discussion includes potential calculation errors and differing methods of approach, which may affect the conclusions drawn. The reliance on specific algebraic manipulations may also introduce assumptions that are not universally accepted.

anemone
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Solve in real numbers the equation $\sqrt[3]{a-1}+\sqrt[3]{a}+\sqrt[3]{a+1}=0$
 
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anemone said:
Solve in real numbers the equation $\sqrt[3]{a-1}+\sqrt[3]{a}+\sqrt[3]{a+1}=0$

using

x+y+z=0=>$x^3+y^3+x^3 = 3xyz$

we get

$(3a)^3 = 3a(a^2-1)$
a= 0 or +/-$\sqrt(1/8)$
 
kaliprasad said:
using

x+y+z=0=>$x^3+y^3+x^3 = 3xyz$

we get

$(3a)^3 = 3a(a^2-1)$
a= 0 or +/-$\sqrt(1/8)$
Neat method, but I get a different answer.

[sp]From $3a = 3\sqrt[3]{a(a^2-1)}$, I get $a^3 = a(a^2-1)$, with $a=0$ the only solution.[/sp]
 
Opalg said:
Neat method, but I get a different answer.

[sp]From $3a = 3\sqrt[3]{a(a^2-1)}$, I get $a^3 = a(a^2-1)$, with $a=0$ the only solution.[/sp]

There was a calculation mistake in my method
 
Thanks to both of you for participating and yes, $a=0$ is the only answer to the problem.:)
 

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